**Subject :**Microwave Engineering

## TM modes in Rectangular Waveguides

**Introduction : -
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In this topic we are going to analyze the propagation of TM modes in a rectangular waveguide.

**Explanation : -
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The TMmn modes in a rectangular guide are characterized by Hz = O. In other words, the z component of an electric field E must exist in order to have energy transmission in the guide. Consequently, the Helmholtz equation for E in the rectangular coordinates is given by :

-------(1)

A solution of the Helmholtz equation is in the form of

............(2)

which must be determined according to the given boundary conditions. The procedures for doing so are similar to those used in finding the TE-mode wave.

The boundary conditions on E_{z} require that the field vanishes at the waveguide walls, since the tangent component of the electric field E_{z }is zero on the conducting surface. This requirement is that for E_{z}= 0 at x = 0, a, then B_{n}= 0, and for E_{z}= 0 at y = 0, b, then D_{n} = 0. Thus the solution as shown in Eq. (2) reduces to

..........(3)

where m = 1, 2, 3, . . ., and n = 1, 2, 3, . . .

If either m = 0 or n = 0, the field intensities all vanish. So there is no TM_{01} or TM_{10} mode in a rectangular waveguide, which means that TE_{10} is the dominant mode in a rectangular waveguide for a > b. For Hz = 0, the field equations, after expanding

are given by

...........(4)

These equations can be solved simultaneously for E_{x} , E_{y} ,H_{x}, and H_{y} in terms of E_{z} .The resultant field equations for TM modes are